📚 Mastering Quadratic Equations for IGCSE Maths | IGCSE 数学二次方程全面突破
Quadratic equations are one of the highest-yield topics in IGCSE Mathematics. They appear in Paper 2 and Paper 4, in pure algebra questions, geometry contexts, and word problems. This article covers every standard method you need: factorising, completing the square, the quadratic formula, and graphical solutions. It also explains the discriminant, common mistakes, and exam-style applications.
二次方程是 IGCSE 数学中分值最高、考查最频繁的主题之一。它们出现在 Paper 2 和 Paper 4 中，可能以纯代数题、几何背景题或应用题形式出现。本文覆盖你需要掌握的所有标准方法：因式分解法、配方法、求根公式法和图像法。文中还讲解判别式、常见错误以及考试风格的应用题。
1. What Is a Quadratic Equation? | 什么是二次方程？
, b, and c are constants and a ≠ 0. The highest power of x is 2, which is why it is called 'quadratic'. If a = 0, the equation becomes linear, not quadratic.
一元二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。x 的最高次数是 2，因此称为“二次”。如果 a = 0，方程就变成一次方程，而不是二次方程。
ions once simplified. The values of x that make the equation true are called the roots or solutions of the equation.
例如，2x² − 5x + 3 = 0、x² = 9 和 (x − 1)(x + 4) = 0 经过化简后都是二次方程。使方程成立的 x 值称为方程的根或解。
2. Solving by Factorising | 因式分解法求解
Factorising is the fastest method when the quadratic expression can be written as a product of two linear brackets. The key idea is: if the product of two terms is zero, then at least one of the terms must be zero. This is called the zero product property.
因式分解法是当二次表达式可以写成两个一次括号的乘积时最快的方法。核心思想是：如果两个因式的乘积为零，那么至少有一个因式必须为零。这称为零乘积性质。
Steps for solving ax² + bx + c = 0 by factorising:
- Rearrange the equation so that one side is 0.
- Factorise the quadratic expression into two brackets.
- Set each bracket equal to 0.
- Solve each linear equation to find the roots.
因式分解法求解 ax² + bx + c = 0 的步骤：
- 重新整理方程，使其中一边为 0。
- 将二次表达式分解成两个括号。
- 令每个括号等于 0。
- 解每个一次方程，得到根。
Example: Solve x² − 7x + 12 = 0.
x² − 7x + 12 = (x − 3)(x − 4) = 0
So x − 3 = 0 or x − 4 = 0, giving x = 3 or x = 4.
示例：解方程 x² − 7x + 12 = 0。
x² − 7x + 12 = (x − 3)(x − 4) = 0
因此 x − 3 = 0 或 x − 4 = 0，得到 x = 3 或 x = 4。
. For example, 2x² + 5x − 3 = 0 factorises as (2x − 1)(x + 3) = 0, giving x = ½ or x = −3.
当 x² 的系数不是 1 时，你必须更仔细地分解，通常使用拆分中间项或试错法。例如，2x² + 5x − 3 = 0 可以分解为 (2x − 1)(x + 3) = 0，得到 x = ½ 或 x = −3。
3. Solving by Completing the Square | 配方法求解
Completing the square is a reliable algebraic method that works for every quadratic equation, even when factorising is difficult or impossible. The method transforms the quadratic expression into a perfect square plus or minus a constant.
配方法是一种可靠的代数方法，对每个二次方程都适用，即使因式分解困难或不可能。该方法将二次表达式转化为完全平方加或减一个常数。
The general idea is to rewrite ax² + bx + c in the form a(x + p)² + q, where p and q are constants. To complete the square for x² + bx, you add and subtract (b/2)²:
一般思路是将 ax² + bx + c 改写成 a(x + p)² + q 的形式，其中 p 和 q 为常数。对 x² + bx 进行配方时，你需要加上并减去 (b/2)²：
x² + bx = (x + b/2)² − (b/2)²
Example: Solve x² + 6x − 7 = 0 by completing the square.
x² + 6x = 7
Add (6/2)² = 9 to both sides:
x² + 6x + 9 = 7 + 9
(x + 3)² = 16
Take the square root of both sides:
x + 3 = ±4
So x = −3 + 4 = 1 or x = −3 − 4 = −7.
示例：用配方法解方程 x² + 6x − 7 = 0。
x² + 6x = 7
两边同时加上 (6/2)² = 9：
x² + 6x + 9 = 7 + 9
(x + 3)² = 16
两边开平方：
x + 3 = ±4
所以 x = −3 + 4 = 1 或 x = −3 − 4 = −7。
d to find the vertex of a quadratic graph, because y = (x + p)² + q has vertex at (−p, q).
当 x² 的系数不是 1 时，先让整个方程除以该系数。配方法还用于求二次图像顶点，因为 y = (x + p)² + q 的顶点位于 (−p, q)。
4. Solving by the Quadratic Formula | 求根公式法求解
The formula is derived from completing the square and is given in your IGCSE formula sheet:
求根公式是一个通用工具，可以直接给出任意二次方程 ax² + bx + c = 0 的根。该公式由配方法推导而来，并在你的 IGCSE 公式表中给出：
x = (−b ± √b² − 4ac) / 2a
To use the formula, first identify the values of a, b, and c from the standard form. Then substitute carefully inside the square root first, then simplify the whole expression.
使用公式时，首先从标准形式中识别 a、b、c 的值。然后先代入根号内部进行计算，再化简整个表达式。
Example: Solve 3x² − 4x − 2 = 0 using the quadratic formula.
Here a = 3, b = −4, c = −2.
x = (4 ± √((−4)² − 4 × 3 × (−2))) / (2 × 3)
x = (4 ± √16 + 24) / 6
x = (4 ± √40) / 6
x = (4 ± 2√10) / 6 = (2 ± √10) / 3
The two roots are x = (2 + √10)/3 and x = (2 − √10)/3.
示例：用求根公式解方程 3x² − 4x − 2 = 0。
这里 a = 3，b = −4，c = −2。
x = (4 ± √((−4)² − 4 × 3 × (−2))) / (2 × 3)
x = (4 ± √16 + 24) / 6
x = (4 ± √40) / 6
x = (4 ± 2√10) / 6 = (2 ± √10) / 3
两个根为 x = (2 + √10)/3 和 x = (2 − √10)/3。
You should leave answers in exact surd form unless the question asks for decimal approximations. The quadratic formula is especially useful when the coefficients are large, negative, or non-integer.
除非题目要求小数近似值，否则答案应保留精确的根式形式。当系数较大、为负数或非整数时，求根公式特别有用。
5. Solving by Graphing | 图像法求解
rcepts. These are the points where the curve crosses the x-axis, so y = 0 at those points.
二次方程可以通过绘制曲线 y = ax² + bx + c 并找到 x 轴截距来求解。这些是曲线与 x 轴相交的点，在这些点上 y = 0。
rdinates of the intersection points with the x-axis are the solutions of ax² + bx + c = 0.
二次函数的图像是一条抛物线。如果 a > 0，开口向上；如果 a < 0，开口向下。与 x 轴交点的 x 坐标就是 ax² + bx + c = 0 的解。
o the solutions are x = −1 and x = 3. This matches the factorised form (x + 1)(x − 3) = 0.
例如，要用图像法解 x² − 2x − 3 = 0，画出曲线 y = x² − 2x − 3。它与 x 轴相交于 x = −1 和 x = 3，因此解为 x = −1 和 x = 3。这与因式分解形式 (x + 1)(x − 3) = 0 一致。
Graphical solutions can be less precise, but they help you understand the number of real roots. If the parabola does not cross the x-axis, the equation has no real roots. If it touches the x-axis at one point, there is one repeated root.
图像解的精度可能较低，但可以帮助你理解实根的数量。如果抛物线不与 x 轴相交，方程没有实根。如果它在一点处与 x 轴相切，则有一个重根。
6. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the part of the quadratic formula under the square root: b² − 4ac. It tells you how many real roots a quadratic equation has without solving the equation fully.
判别式是求根公式中平方根下的部分：b² − 4ac。它告诉你二次方程有多少个实根，而无需完整解方程。
The three cases for the discriminant are:
- If b² − 4ac > 0, the equation has two distinct real roots.
- If b² − 4ac = 0, the equation has one repeated real root (or two equal roots).
- If b² − 4ac < 0, the equation has no real roots (the roots are complex, which you do not need for IGCSE).
判别式的三种情况如下：
- 如果 b² − 4ac > 0，方程有两个不同的实根。
- 如果 b² − 4ac = 0，方程有一个重复的实根（或两个相等的根）。
- 如果 b² − 4ac < 0，方程没有实根（根为复数，IGCSE 不要求）。
has no real roots. This also means the graph of y = 2x² − 4x + 5 never crosses the x-axis.
示例：对于方程 2x² − 4x + 5 = 0，判别式为 (−4)² − 4 × 2 × 5 = 16 − 40 = −24。由于判别式为负，方程没有实根。这也意味着 y = 2x² − 4x + 5 的图像从不与 x 轴相交。
d root, or no real roots. Set up the inequality b² − 4ac > 0, = 0, or < 0 and solve for k.
考试题经常要求你求参数 k 的取值范围，使得二次方程有两个不同实根、一个重根或无实根。建立不等式 b² − 4ac > 0、= 0 或 < 0，然后解出 k。
7. Quadratic Equations in Disguise | 隐藏的二次方程
Some equations are not initially in quadratic form, but can be rearranged by substitution to become quadratic. Common examples include equations with x⁴, x⁻², or square roots.
有些方程最初不是二次形式，但可以通过代换重新整理成二次方程。常见例子包括含有 x⁴、x⁻² 或平方根的方程。
4) = 0, so y = 1 or y = 4. Since y = x², we get x² = 1 or x² = 4, giving x = ±1 or x = ±2.
示例：解方程 x⁴ − 5x² + 4 = 0。令 y = x²，方程变为 y² − 5y + 4 = 0。因式分解得到 (y − 1)(y − 4) = 0，所以 y = 1 或 y = 4。由于 y = x²，得到 x² = 1 或 x² = 4，因此 x = ±1 或 x = ±2。
get x² + 1 = 5x, then rearrange to x² − 5x + 1 = 0 and solve using the quadratic formula.
另一种常见类型是分式方程，在去分母后化为二次方程，例如 x + 1/x = 5。两边乘以 x 得到 x² + 1 = 5x，再整理为 x² − 5x + 1 = 0，然后用求根公式求解。
Always check for extraneous solutions when you multiply by a variable expression, especially if the variable could be zero. IGCSE examiners often include such traps.
当你乘以含变量的表达式时，一定要检查是否有增根，尤其是变量可能为零时。IGCSE 考官经常设置这类陷阱。
8. Quadratic Word Problems | 二次方程应用题
Word problems require you to translate a real-life scenario into a quadratic equation, then solve it and interpret the answer in context. Common contexts include area, projectile motion, consecutive numbers, and profit.
应用题要求你将实际情景转化为二次方程，然后求解并在情境中解释答案。常见情境包括面积、抛体运动、连续整数和利润。
Example: The length of a rectangle is 3 cm longer than its width. The area is 40 cm². Find the dimensions of the rectangle.
= 0, so x = −8 or x = 5. Since width cannot be negative, x = 5 cm, and the length is 8 cm.
示例：一个矩形的长比宽长 3 cm。面积为 40 cm²。求矩形的尺寸。
设宽为 x cm，长为 x + 3 cm。面积 = 长 × 宽，所以 x(x + 3) = 40。得到 x² + 3x − 40 = 0。因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于宽度不能为负，x = 5 cm，长为 8 cm。
Always reject solutions that are not physically meaningful, such as negative lengths, times, or quantities. Also check that your answer satisfies the original conditions of the problem, not just the equation.
始终排除物理上无意义的解，例如负长度、负时间或负数量。同时检查你的答案是否满足题目的原始条件，而不仅仅是方程。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
roduct property only works when the product is zero. You must first write x² − 5x − 6 = 0.
一个常见错误是在因式分解前忘记把方程整理为等于零。例如，解 x² − 5x = 6 时写成 x(x − 5) = 6 是无效的，因为零乘积性质只在乘积为零时成立。你必须先写成 x² − 5x − 6 = 0。
Another common error is mishandling the minus signs in the quadratic formula, especially when b is negative. Write the formula with parentheses around b and the discriminant to avoid sign mistakes. Also simplify the square root fully if possible.
另一个常见错误是在求根公式中处理负号不当，尤其是当 b 为负时。在 b 和判别式周围加上括号以避免符号错误。如果可能，还要将平方根完全化简。
When completing the square, remember to add the same value to both sides of the equation. Do not just add (b/2)² to the left side. If the coefficient of x² is not 1, divide first. Also, when taking square roots, include both the positive and negative roots.
配方法时，记住要在方程两边加上相同的值。不要只在左边加 (b/2)²。如果 x² 的系数不是 1，先除以该系数。另外，开平方时，要包括正负两个根。
For graphical questions, draw the curve accurately enough to read off the roots, or use a table of values. Always label the axes and mark the x-intercepts clearly. If you use a calculator to find roots, show the substitution into the quadratic formula in your working.
对于图像题，曲线要画得足够准确以读出根，或使用数值表。始终标注坐标轴并清楚标出 x 轴截距。如果使用计算器求根，要在解题过程中展示代入求根公式的步骤。
10. Practice Questions and Summary | 练习题与总结
Here are some IGCSE-style questions to test your understanding. Try them before looking at the answers.
以下是一些 IGCSE 风格的题目，用来检验你的理解。先尝试作答，再看答案。
- Solve x² − 5x + 6 = 0 by factorising.
- Solve 2x² + 3x − 2 = 0 using the quadratic formula.
- Solve x² − 8x + 5 = 0 by completing the square, giving your answer in surd form.
- Find the values of k for which the equation x² + kx + 4 = 0 has exactly one real root.
- The product of two consecutive positive integers is 56. Find the integers.
- 用因式分解法解 x² − 5x + 6 = 0。
- 用求根公式解 2x² + 3x − 2 = 0。
- 用配方法解 x² − 8x + 5 = 0，答案用根式形式表示。
- 求 k 的值，使得方程 x² + kx + 4 = 0 恰好有一个实根。
- 两个连续正整数的乘积为 56。求这两个整数。
Summary of key points:
- Quadratic equations have the form ax² + bx + c = 0 with a ≠ 0.
- Four solving methods: factorising, completing the square, quadratic formula, and graphing.
- The discriminant b² − 4ac determines the number of real roots.
- Always check solutions in context and reject extraneous answers.
关键点总结：
- 二次方程的形式为 ax² + bx + c = 0，且 a ≠ 0。
- 四种求解方法：因式分解法、配方法、求根公式法和图像法。
- 判别式 b² − 4ac 决定实根的个数。
- 始终在情境中检查解，并排除增根。
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